## Abstract

A computational model is presented to nd the q-Bernstein quasi-minimal Bezier surfaces as the extremal of Dirichlet functional,

and the Bezier surfaces are used quite frequently in the literature of computer science for computer graphics and the related

disciplines. The recent work [1–5] on q-Bernstein–Bezier surfaces leads the way to the new generalizations of ´ q-Bernstein

polynomial Bezier surfaces for the related Plateau–Bézier problem. The q-Bernstein polynomial-based Plateau–Bézier problem is

the minimal area surface amongst all the q-Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. ´

Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related

Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the

corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal conditions for which

the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the q-Bernstein

polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet

functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be

solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is q-Bernstein–Bézier

minimal surface

and the Bezier surfaces are used quite frequently in the literature of computer science for computer graphics and the related

disciplines. The recent work [1–5] on q-Bernstein–Bezier surfaces leads the way to the new generalizations of ´ q-Bernstein

polynomial Bezier surfaces for the related Plateau–Bézier problem. The q-Bernstein polynomial-based Plateau–Bézier problem is

the minimal area surface amongst all the q-Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. ´

Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related

Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the

corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal conditions for which

the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the q-Bernstein

polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet

functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be

solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is q-Bernstein–Bézier

minimal surface

Original language | English |
---|---|

Article number | 8994112 |

Number of pages | 21 |

Journal | Journal of Mathematics |

Volume | 2022 |

Publication status | Published - 22 Sept 2022 |

## Keywords

- Bezier surfaces
- q-Bernstein polynomial-based Plateau–Bezier problem
- Dirichlet functional
- q-Bernstein–Bezier minimal surface
- optimization theory